C₂×#2760 #52198

order	256
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	Aut(G) order 3145728
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#2760

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵
prime radical	C₂×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1467 conjugacy classes of subgroups; only the 463 normal subgroups are shown.

normal subgroups only

This group has 1467 conjugacy classes of subgroups — too many to draw. Only its 463 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 26 points:

(1,2);(4,7)(5,9)(11,12,15,13)(14,17,18,16)(19,20)(21,25)(22,23)(24,26);(4,7)(8,10)(12,13)(14,18)(20,23)(21,26);(3,4,6,7)(5,8,9,10)(11,13,15,12)(14,17,18,16)(19,21)(20,24)(22,26)(23,25);(3,5)(4,8)(6,9)(7,10)(11,14)(12,16)(13,17)(15,18)(19,20)(21,24)(22,23)(25,26)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#2760⋊C₃, #72231952
maximal subgroup of	#131604, C₂×#12187, C₂×#12263, …
maximal normal subgroup of	#131604, C₂×#12187, C₂×#12263, …
maximal quotient of	C₂×#14375, C₂×#14508, C₂×#14572, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45186);